Solved – Order statistics of equal correlated continuous random variables

Suppose that $X_1, ldots, X_n$ are mutlivariate normal with equal correlation
$rho$ and each of them are marginally
distributed as $N(0,1)$. Let $X_{(1)}, ldots, X_{(n)}$
be the corresponding order statistics. The distribution of
$X_{(1)}$ and $X_{(n)}$ are easily found. What about the
distribution of the other order statistics? Can anyone
give some reference on this?

Thank you. Hanna

As shown in Tong, Y. L. (1990). Multivariate normal distribution. Springer-Verlag., ch. 6, for the setup described in the question and for non-negative correlation coefficient $rhoin [0,;1)$, the distribution function (cdf) and the density of an order statistic $X_{(i)}$ are (where $phi()$ and $Phi()$ are the standard normal pdf and cdf)

$$G_{(i)}(x) = int_{-infty}^{infty}F_{(i)}left(frac{x+sqrt{rho}z}{sqrt{1-rho}}right)phi(z)dz$$

and differentiating,

$$g_{(i)}(x) = int_{-infty}^{infty}frac 1{sqrt{1-rho}}f_{(i)}left(frac{x+sqrt{rho}z}{sqrt{1-rho}}right)phi(z)dz$$


$$f_{(i)}(y) = frac{n!}{(i-1)!(n-i)!}[Phi(y)]^{i-1}[Phi(-y)]^{n-i}phi(y)$$ and $$F_{(i)}(y) = sum_{j=i}^n {n choose j}[Phi(y)]^{j}[Phi(-y)]^{n-j}$$

i.e $f_{(i)}(y)$ and $F_{(i)}(y)$ are the pdf and cdf of the order statistic $(i)$ from an i.i.d. standard normal random sample.

For the corresponding results when the correlation coefficient is negative, the author refers to the book "Order Statistics", by H.A. David & H.N. Nagaraja ch. 5 (now in its 3d edition, 2003).

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